Journal of the ACM (JACM)
Bounds to Complexities of Networks for Sorting and for Switching
Journal of the ACM (JACM)
Shifting Graphs and Their Applications
Journal of the ACM (JACM)
The complexity of monotone functions.
The complexity of monotone functions.
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In this paper we study the complexity of realizing a monotone but otherwise arbitrary Boolean function. We consider realizations by means of networks and formulae. In both cases the possibility exists that although a monotone function can always be realized in terms of monotone basis functions, a more economical realization may be possible if basis functions that are not themselves monotone are used. Thus, we have four cases, namely: 1. The cost of realizing a monotone function with a network over a universal basis. 2. The cost of realizing a monotone function with a network over a monotone basis. 3. The cost of realizing a monotone function with a formula over a universal basis. 4. The cost of realizing a monotone function with a formula over a monotone basis. For the first case, we obtain a complete solution to the problem. For the other three cases, we obtain improvements over previous results and come within a logarithmic factor or two of a complete solution.